Tree Automata MuCalculus and Determinacy

Extended Abstract

EA Emerson and CS Jutla

The University of Texas at Austin

IBM TJ Watson Research Center

Abstract that tree automata are closed under disjunction pro

jection and complementation While the rst two

We show that the prop ositional MuCalculus is eq

are rather easy the pro of of Rabins Complemen

uivalent in expressivepower to nite automata on in

tation Lemma is extraordinarily complex and di

nite trees Since complementation is trivial in the Mu

cult Because of the imp ortance of the Complemen

Calculus our equivalence provides a radically sim

tation Lemma a numb er of authors have endeavored

plied alternative pro of of Rabins complementation

and continue to endeavor to simplify the argument

lemma for tree automata which is the heart of one

HR GH MS Mu Perhaps the b est

of the deep est decidability results We also showhow

known of these is the imp ortant work of Gurevich

MuCalculus can b e used to establish determinacy of

and Harrington GH which attacks the problem

innite games used in earlier pro ofs of complementa

from the standp oint of determinacy of innite games

tion lemma and certain games used in the theory of

While the presentation is brief the argument is still

online algorithms

extremely dicult and is probably b est appreciated

Intro duction

y the page supplementofMonk when accompanied b

Mon

We prop ose the prop ositional Mucalculus as a

uniform framework for understanding and simplify

In this pap er we present a new enormously sim

ing the imp ortant and technically challenging areas

plied pro of of the Complementation Lemma We

of automata on innite trees and determinacy of in

would argue that the new pro of is straightforward and

nite games We show that the MuCalculus is pre

natural To obtain our pro of weshow that the prop o

cisely equivalent to tree automata in expressivepower

sitional MuCalculus Ko is expressively equiva

p ermitting a radically simplied pro of of the Comple

lent to nondeterministic tree automata Since the

mentation Lemma for tree automata We also show

MuCalculus is trivially closed under complementa

how to systematically prove the determinacy of cer

tion the equivalence immediately implies that tree

tain innite games including games in the theory of

automata are also closed under complementation

online algorithms

We remark that the equivalence of the Mucalculus

Rabin intro duced tree automata nite state

and tree automata is a new result of some indep en

automata on innite trees to prove the decidabil

dentinterest since it shows that there indeed exists a

ity of the monadic second order theory of n succes

natural mo dal logic of programs precisely equivalent

sors SnS This is one of the most fundamental de

to tree automata in expressive power cf Th

cidability results to which many other decidability

Actually a brief argument establishing this equiva

results in logic mathematics and computer science

lence can be given by app ealing to known lengthy

can be reduced The pro of involves reducing satis

arguments translating through SnS However a care

ability of SnS formulae to the nonemptiness prob

ful examination of this argumentreveals that rep eated

lem of tree automata The reduction entails showing

app eals to the Complementation Lemma are involved

Thus this argument is not at all suitable for use in a us to avoid the app eal to GH in our translation

pro of of the Complementation Lemma The relation That proves the equivalence of nondeterministic Tree

ship of the MuCalculus and tree automata has also Automata and the MuCalculus and hence the Com

been investigated previously by Niwinski In Ni plementation Lemma

it is shown that tree automata can b e translated into

Almost all published pro ofs of the Complementa

the MuCalculus while in Ni it is shown that a re

tion Lemma RaBu GH can be seen

stricted MuCalculus in which conjunctions are not

up on reection to havetaken the following approach

allowed can b e translated into tree automata As we

b est broughtoutby MS

shall see in our translation of the full MuCalculus

Alternating Tree Automata are easy to comple

into tree automata the main diculty lies in dealing

ment given the fact that certain innite games

with the conjuncts

are determinedie at least one of the play ers has a

The MuCalculus L is a mo dal logic we deal winning strategy Nondeterministic Tree Automata

here with the prop ositional Temp oral version with clearly are alternating Tree Automata Alternating

xp oint op erators It provides a least xpoint operator Tree Automata can be reduced to nondeterministic

and a greatest xpoint operator which makes Automata as shown in this pap er using determiniza

it p ossible to give extremal xp ointcharacterizations tion of Automata and aforgetful strategy theo

of the branching time mo dalities Intuitiv rem ely the Mu

Calculus makes it p ossible to characterize the mo dal

Theorems and in all the results mentioned

ities in terms of recursively dened treelike patterns

ab ove have had dicult pro ofs Although GH

First we show that the parse tree of a formula of

have somewhat simplied the pro ofs over Ra the

L can b e viewed as an alternating Tree Automaton

pro ofs are far from b eing transparent Moreover their

MS giving a translation from L to alternating

pro ofs of Theorem and are intertwined The

Tree Automata The problem of translating the Mu

new pro of of the Complementation Lemma via Mu

Calculus into nondeterministic Tree Automata then

Calculus suggests the following for simplifying the

reduces to translating alternating Tree Automata to

ab ove approach ie and as well To prov e

nondeterministic Tree Automata

extremal xp oints Mucalculus can b e used to

The alternating Tree Automata obtained from the characterize games in which a particular Player has a

MuCalculus however have a nice prop erty in the winning strategy Indeed in section we give simple

sense that if there is an accepting run of an automa and straightforward Mucalculus characterizations of

ton A on an input tree t then there is a historyfree games in which Players I and I I have winning strate

accepting run of A on t A run of A on t is historyfree gies which turn out to b e trivial complements of each

i the nondeterministic choices made by A along any other Moreover a totally indep endent ranking argu

t state ie the path of t dep end only on the curren mentprovides a simple pro of of

choices are indep endent of the history of the dierent

Finallywealsoshow that the ab ovetechnique of

forallruns

extremal xp oint Mucalculus characterization gives

Weshowhow these historyfree alternating Tree a general metho d for proving various determinacy re

Automata can be reduced to nondeterministic Tree sults Wolfe cf Mo proved the deter

Automata using constructions for determinizingSa minacy of innite games in the second level of the

and complementing EJ automata on innite strings Borel hierarchyF Weshowthatitispossibleto

For general alternating tree automata we can still do view such determinacy results as implicitly construct

the translation into nondeterministic tree automata ing extremal xp oints MuCalculus characterization

but nowwemust use Gurevich and Harringtons for also gives a clear and simple pro of of determinacy of

getful strategy theorem GH the historyfreedom sp ecialized F games used in derandomizing exis

of the automata derived from the MuCalculus allows tentially Online algorithms RS This makes the

sequence of OR no des s s s visited along b in the result in RS only slightly more complicated b eing

 k 

play which we dene by a function path Strat more general than a determinacy result in B

I

f g OR

Detailed pro ofs of Theorems in section and

Similarly given app ear in the second authors dissertation J b f g and a legal strategy

there is also a unique innite sequence of OR

I

Alternating Tree Automata to Nondeter

no des visited along b if Player I follows Thus

I

ministic Tree Automata

we dene a function ipath Strat f g OR

I

We deal here only with nite state Automata

with ipath bs s OR where b b b

I  

such that s b and k s

Denition A nondeterministic binary TreeAu

 I k

 

b b b b Note that fpath b jb pre

tomaton is a tuple N OR AN D

I  k  k  I

x of bg is exactly the set of nonempty nite prexes

Ls where

of ipath b

is the input alphab et

I

OR is the nite set of states

The acceptance condition is given by a sub

AN D is a set of no des intuitively corresp onding to

set of OR We say that x OR satises i

entries in the transition function of the Automaton

x is in the subset representing Usually is de

L AN D is a lab elling of the AN D no des with

ned by a Temp oral formula interpretted over in

the input symb ols

nite sequences of OR no des For example the Streett

is a relation on OR AN D dening the dierent

acceptance condition has m pairs of subsets of OR

nondeterministic choices from OR to AN D no des es

fGR E E N RED GR E E N RED g Then

 i m m

V

sentially the transition function

the subset of OR dening is given by GF

im

AN D f g OR is a function sp ecifying left

GR E E N GF R E D In Temporal Logic G stands

i i

and right successor states

for everywhere and F stands for eventually The

s OR is the start state

Pairs acceptance condition is the complementofthe

is the acceptance condition to b e dened b elow

Streett condition

Wenowgive a game theoretic denition of acceptance

Denition A legal strategy forIisawinning

I

of a tree t by an Automaton N

strategy for I i

b f g ipath b satises

Denition Given an input tree t f g

I

Wesaythat N accepts t i I has a winning strategy in

wecandeneaplayer innite game N tonthe

N t LN ftjNaccepts tree tg is the language

ab ove bipartite graph as follows Player I picks AN D

accepted by N

successors of OR no des picked by Player I I and vice

versa

Nondeterministic Automata and TMs were gener

alized to Alternating Automata and TMs in CKS

Formally a strategy for Player I is a map

I

Alternating tree Automata were rst dened in MS

f g AN D with the requirementthat x f g

The following denition of Alternating Tree Automa

L x tx We say that is a legal strat

I I

k 

ton is the generalization of the denition of nondeter

egy for I i s and b b b f g

I  k

ministic Tree Automaton dened ab ove to include

b b b b b b b We need the

I  k  k I  k

universal states Intuitivelywenowhavetwo kinds

notion of legal to ensure that I indeed picks AN D

of AN D no des DAN D and AAN D The DAN D

no des which are successors of previously picked OR

no des are as b efore the directional AN D no des with

no des The class of legal strategies of Player I will

two successors sp ecifying the transitions in the left

be called Strat A strategy for Player II is a map

I



and right directions The AAN D no des are the dual

AN D f g dened similarly

II

of the OR no des representing the nondeterminism

In a play given b b b b ie the moves of

 k

utoma Denition An Alternating binary TreeA

Player I I and a legal strategy there is a unique I

ton is a tuple A OR DAN D condition

AAN D Ls where

Given b f g and a legal strategy wenow

I

is the input alphab et

dene a function bundl e which gives the set of se

OR is the nite set of states

quences of OR no des Each sequence in the bun

DAN D is a set of no des with successors sp ecifying

dle will be referred to as a thr ead Thus bundl e

the transitions in the dierent directions

OR

AS tr at f g such that

I

AAN D is a set of no des with successors sp ecifying

bundl e fs g

I

the dierent nondeterministic transitions

bundl e b fs s j s b s g

I  I 

OR DAN D denes the dierent nondetermi

bundl e b b b fs s s s j s s s

I  k  k k   k

nistic choices from the OR no des

bundl eb b and s b b s b s g

k  I k  k k k 

DAN D f g AAN D is a function dening

Similarlywe dene a function ibundl e AS tr at

I

the transitions along the dierent directions

OR

f g suchthatibundl e b b

I 

AAN D OR denes the dierent nondeter

fs s jk s s s bundl e b b b g

  k I  k 

ministic transitions

b will be referred to as Each element of ibundl e

I

L DAN D is a lab elling of DAN D no des with

ithr ead Note that ibundl e has Automaton A and

input symbols

tree t as implicit arguments

s OR is the start state

is the Acceptance Condition to b e dened later

As for Nondeterministic Tree Automata the ac

ceptance condition is given by a subset of OR

Wenowgive a game theoretic denition of accep

Wesay that x OR satises i x is in the subset

tance of a tree t by an Automaton A

representing Once again wemay sp ecify using

Denition Given t f g a player in

Temp oral Logic

nite game A t can b e dened as for Nondetermi

Denition A legal strategy for Player I is a

I

nistic Automata Player I picks a no de from DAN D

winning strategy for Player I i

while Player I I picks a pair a binary digit dening an

b f g s ibundl e b s satises

I

AAN D no de and then an OR no de successor of the

Wesay that A accepts t i I has a winning strategy in

resulting AAN D no de Thus Player I I has additional

At LA ftjAaccepts tree tg is the language

choice It not only picks a path b f g it also

accepted by A

picks a nondeterministic sequence of transitions

Denition A strategy for Player I in a game

I

Thus a strategy for Player I is a map fs g

I

AtisaHistoryfreeStrategy i at eachnodeofthe

f g OR DAN D with the requirement that

input tree the strategy dep ends only on the current

bs f g OR L s bs tb Wesay

I

state of the Automaton A and the no de in the tree t

that is legal i bs f g OR s s

I k I

Formallyi

bs where s s s The class of strategies of

 k

k k  

k b f g s s s s OR s

 k

Player I will b e called AS tr at A strategy for player

I

   k   

s s s OR s s sb s b



k I I



k k

f g OR II isamap DAN D

II

An Alternating Automaton A is a Historyfree Al

In contrast to the games for Nondeterministic Auto

ternating Automaton i for every tree t if Player I

mata given b f g and a legal strategy there

I

has a winning strategy in A t then Player I has a

is a set of sequences of OR no des rather than a unique

historyfree winning strategy in At

sequence of OR no des The dierent sequences in the

Theorem GH If Player I has a winning strat

set corresp ond to dierent nondeterministic choices

egy in At then it has a forgetful winning strategy

of Player I I Player Is strategy now is a winning

I

in At ie a strategy which only dep ends on a

strategy i for all b f g all the resulting innite

small nite history of the play where A is dened

sequences of OR no des along b satisfy the acceptance

with Muller Acceptance condition

translate Prop ositional MuCalculus to Rabin Tree Construction

Automata We begin by dening the Temp oral ver

Given an Alternating Automaton A the obligation

sion of Prop ositional MuCalculus Following that

is to construct a nondeterministic Tree Automaton

we convert a MuCalculus formula to an equivalent

N such that for every tree tPlayer I has a winning

Streett historyfree Alternating Tree Automaton

strategy in A tiPlayer I has a winning strategy

Prop ositional MuCalculus in N t A legal strategy in A t generates for

I

each b f g anibundl e bofinnite ithr eads

I

Denition The formulae of the Prop ositional

whereas a legal strategy in N t generates for

I

MuCalculus are

each b f g a single thread ipath b

I

Prop ositional letters P Q R

Thus N must collect all the ithr eadsinibundl e Prop ositional variables XYZ

into a single ipath and if ibundl e p p q and p q wherep and q are anyformulae is generated bya

EX p and AX pwherep is any formula nite state mechanism then the ipath must also be

X f X and Xf X where f X is any for generated by a nite state mechanism Safras con

mula syntactically monotone in the prop ositional vari struction Sa eg is a mechanism which generates

able X a single ipath satisfying Streett acceptance condition

i all the ithr eads satisfy the complement of the Buchi

A sentence is a formula containing no free prop o

acceptance condition A mo dication of the coSafra

sitional variables In the sequel we will use as a

Construction EJ generates a single ipath satisfy

generic symb ol for or Sentences are interpreted

ing pairs acceptance condition i all ithr eads satisfy

in KripkeTrees

the Streett acceptance condition which is what we

Denition A Kripke Tree is an innite binary

require here

tree t f g Prop A satisfaction relation jis

There are two main points to be noted Firstly

dened b etween f g and Prop Wesay x j P i

N must at each OR state have a transition function

P tx Note that usually a Kripke tree is dened

which is the cross pro duct of the transition functions

Prop

as a map t f g but since wewant to show

of all the last no des of the thr eads it is collecting into

equivalence of the MuCalculus to Tree Automata we

asinglepath Secondly since the number of thr eads

use the lab elling as in Tree Automata

increase arbitrarily with increasing length if N were

Denition A model is a KripkeTree with the

to collect all the ithr eads N would require innitely

Satisfaction relation extended to all sentences by means

many states Howev er by Theorem every Streett

of the usual b o olean rules and

Alternating tree Automaton is forgetful and hence

x j EX p i i f g x i j p

N only needs to keep the small nite history of the

x j AX p i i f g x i j p

thr eads thus requiring only nitely many states Of

T

x j Xf X i x fS U jS fy jy j f X

course for historyfree Alternating Automaton wedo

with X interpreted as sgg

not need theorem

S

x j Xf X i x fS U jS fy jy j f X

Theorem Given A O R D A AA L

with X interpreted as sgg

s a Streett alternating Tree Automaton we

Every formula has a positive normal form in which

construct an equivalent nondeterministic Pairs Tree

all negations apply directly to prop osition letters De

     

Automaton N OR AN D L s

p to b e the p ositive normal form of p ne not

and ie LA LN If jORj n jDAj m

O nlog n

 k  

has k pairs then jOR j jAN D j

Denition The Fischer Ladner Closure of a

O nlog n O nlog n

k   O nlog n

m withhaving k pairs

sentence p in p ositive normal form is the smallest

set FLp of sentences satisfying the following con

Prop ositional MuCalculus to Tree Automata

straints

Using the Construction in the previous section we

p FLp choice function in StE determines a derivation

if q FLpthen notq FLp tences in the pre relation between o ccurences of sen

if q r FLpthen q r FLp mo del so obtained StE Wewould liketosay that

if q r FLpthen q r FLp a premo del is in fact a mo del when there is no innite

if EX q FLp then q FLp derivation sequence which rederives a musentence in

if AX p FLp then q FLp nitely often However StE show that this claim

if X f X FLp then f X f X FLp is true only when restricted to derivations in which

Xf X FLpthenf Xf X FLp if the given sentences app ear as a subsentence of every

derivation step We saythat a sentence X f X

Denition A premodel is a KripkeTree with a

is regenerated from x to y if X f X at x derives

satisfaction relation j extended to FLp under the

X f X at y in suchaway that X f X is a sub

following constraints

sentence of every derivation step A winning strategy

x j p i x j notp

in the ab ove game is a wel lfounded winning strat

I

x j p q i either x j p or x j q

egy when the regeneration relations for sentences

x j EX p i i f g x i j p

are wellfounded The main theorem in StE states

x j X f X i x j f X f X

that a premo del is a Model i there is a wellfounded

A premo del is almost a mo del except rule p er

winning strategy generating it from the transition dia

mits xf X to be interpreted as an arbitrary x

gram Moreover a wellfounded winning strategy de

point

nes a straightforward historyfree wellfounded win

ning stategy b ecause Player I can pickthechoice with

Construction

the least rank with resp ect to the the regeneration re

Theorem Given a MuCalculus formula f we

lations for a formal pro of see StE

build a Streett Historyfree Alternating Tree Automa

For a xp oint formula X f X we say that X

ton equivalentto f in the sense that the Automaton

is b ound to this formula Wlog assume that in f

accepts exactly those binary trees lab elled with Prop

every xp oint sub expression has a unique X b ound to

which are mo dels of f Here Prop is the set of Prop o

it Wesay that a xp oint sub expression is of higher

sition symb ols in f

precedence than another if the latter is contained as a

Proof Sketch Consider the parse tree of f It can

strict sub expression in the rst For each xp oint sub

be viewed as an Alternating Automatons transition

formula there is a xp ointsentence in FLf given

diagram The no des in the parse tree corresp ond

inductively as follows For xp oint subformulae of

to AAN D no des the no des to OR no des while

maximal precedence the subformulae which are sen

AX p and EX p corresp ond to DAN D no des More

tences in this case itself are in FLf We dene a

over an o ccurence of variable x b ound in xf x is

and onto map H between variables and the x

identied with the no de xf x ie makes a lo op

point sentences o ccuring in FLf corresp onding to

The ab ove transition diagram with acceptance condi

the xp oint sub expression to which the variable is

tion tr ue denes an Alternating Automaton A

b ound Thus as already stated H X f where

Note that the transition diagram dened ab oveisnot

f is a maximal precedence sub expression such that X

necessarily a tripartite graph

is bound to it H X X f X H Y H Y

 k

Given a map fs g ff g ORg DAN D

I

where X is b ound to X f X Y Y inf Clearly

 k

a premo del T can b e generated from the Alter

I

Y are of higher precedence than X and hence the

i

nating Automaton A It can b e shown that T is

I

ab ove denition is wellfounded

a premo del of f extending the lab elling of t i is

I

It is a simple exercise to note that any rederiva

a winning strategy for Player I in At

tion of H X ats to H X att is a regeneration i

Moreover the winning strategy also called the

there is no H Y derived inbetween such that Y is

of higher precedence than X This motivates the fol in all earlier pro ofs as outlined in the Intro duction

lowing mo dication in the acceptance condition of the ever our pro of did not seem to involve dicult How

Alternating Automaton A obtained ab ove suchthat pro ofs of and whereas all known pro ofs of

in the new Streett Alternating Automaton B a win and have b een intimidating This suggests that

ning strategy is a wellfounded winning strategy in A their mightbesimpleproofsof and Indeed

and vice versa Thus the input tree t is a mo del ie we now give simple pro ofs of determinacy of in

has an extendable satisfaction relation as in Denition nite games for Tree Automata and historyfree

i B has a winning strategy in B t Since the strategies for players in such games

only change is in the acceptance condition a history

First note that in the Pro of of theorem in

free strategy in A t remains a historyfree strategy

stead of translating MuCalculus to Streett Automata

in B t Thus B is a historyfree Alternating Au

the natural Automata to which MuCalculus trans

tomaton

lates easily is the one with the following parity ac

Consider the partial order onthesetofvari ceptance condition

ables in f given by the precedence relation Wenow

Denition Let the states of a Tree Automaton

assign an integer heig htX to each X All leaf vari

or the OR no des b e lab elled with colors m For

ables ie Y Y X X in the partial or

an innite path or sequence of no des

der have heig htX Otherwise heig htX

ev en largest io o ccurring color index among k

k

maxheig htX X X The set GR E E N

i i i

is even

is the set of no des corresp onding to X such that

odd largest io o ccurring color index among k

k

heig htX i and X is b ound to a sub expression

is o dd

The set RE D is the set of no des corresp onding to

i

Wesay that x OR satises the parity condition i

X suchthat heig htX i and X is b ound to a

x satises ev en

m

sub expression The Streett acceptance condition is

Stated in terms of the usual Green Red pairs

given by i GREEN io RE D io

i i

the parity condition is i io Gr een and j i

i

It can b e shown that A has a wellfounded winning

fo Red Note that as opp osed to Streett and

j

strategy i B has a winning strategy Thus B is the

Pairs acceptance condition parity acceptance con

required Streett historyfree Alternating Automaton

dition is trivially closed under complementation It

is this prop ertywhich makes proving determinacy of

Tree Complementation Lemma games with parity acceptance condition much easier

In fact a HossleyRacko like nite mo del theorem

From Theorem and Theorem we conclude

for ParityTree Automata also turns out to b e much

that Prop ositional Mucalculus can be translated to

simpler Moreover ParityTree Automata are trivially

Nondeterministic Rabin Pairs Tree Automata More

Automata Also a simple convertable to Pairs Tree

over Niwinski Ni had shown that Nondeterministic

conversion from Pairs Tree Automata to ParityTree

Pairs Automata can be translated to Prop ositional

Automata can be obtained by a slight mo dication

MuCalculus Thus Prop ositional MuCalculus and

of Sa whichconverts a deterministic Pairs Auto

Nondeterministic Tree Automata are expressively eq

mata to deterministic Streett Automata This is also

uivalent Rabins Tree Complementation Lemma ie

the essence of the LAR argument in GH Sim

languages accepted byTree Automata are closed un

ilarly innite games with Muller or Pairs condition

der complementation follows by trivial complemen

easily reduce to Innite games with Parity conditions

tation of MuCalculus

Note that our pro of of complementation via Mu

Calculus did indeed involve Alternating Tree Auto

mata which was also implicitly or explicitly used

For k it holds b ecause y I Determinacy of innite games with parity t col or

n n

winning conditions E X AX y Supp ose it holds for k then for paths in

where s s s s i s is col or and

 i k  i n

We studied a number of dierent games in the

at s E X AX y there is achoice extension of

k 

previous sections However in general a two player

such that all successors y and hence I t col or

n n

innite game can be given by a game tree which is

E X AX y A trivial insp ection shows that along all

an ANDOR innite tree with its no des lab elled with

extensions of the path one of the three disjuncts holds

colors m Wlog we assume that eachORnode

with k incremented and we are done

has the same color as all its AND successors We gen

eralize the ab ove game trees to syntactic game trees If n is o dd to prove I t is the least xp oint let

n

in which no des could be lab elled with MuCalculus b e a strategy for player I in I t st along all

n

expressions with or without free variables mo dality paths of ev en Ft Then the relation R given

n

F stands for eventually A strategy for player I picks as follows on the no des of is wellfounded uR v i

an AND successor at each OR no de while a there is a path from u to v andu or v is lab elled n for strat

otherwise wehave an innite path in with innitely egy for Player I I picks an OR successor at each AND

many n Let u be a R minimal no de in such that no de A strategy in a game tree denes another tree

the game tree starting at u is not in y By virtue of whichwe will call from the given game tree

u is in I t v where v is a no de lab elled col or

n n

Denition

and has successors v v v Note that for every

 i r

I t Set of game trees st Player I has a strategy

k



i uR v since v is lab elled col or No w either for

i n

with all paths satisfying ev en or Ft

k

all suchnodesv if any for all i v is in y in which

i

I tt



case u is in y by Else we found a v not in y

i

II t Set of game trees st Player I I has a strat

k



and uR v whichcontradicts the R minimalityofu

i

egy with all paths satisfying odd or Ft

k

Thus every u in is in y

if n is even and if n is o dd

W

n

Theorem II t x x x

n n n 

W

in

Theorem I t x x x col or

n n n  i

in

col or AX E X x t

i i

E X AX x t

i

Proof An argument symmetric to Theorem

Proof We proveby induction over n Consider the

Corrollary Innite games dened over game

equation

W

trees with no des lab elled with colors m and with

y x x col or E X AX x

n n i i

in

parity winning or acceptance conditions are deter

col or EX AXy t when n

n

mined

and y t when n

History free winning strategies

Using induction hyp othesis on the rhs the equa

tion ab ove b ecomes Weshow that if a player has a winning strategy in

y I t col or E X AX y a parity innite game then the player has a winning

n n

strategy which do es not dep end on the history More

I t is easily seen to b e the xp oint of the ab ove

n

precisely a Player Is winning strategy is historyfree

equation y I t col or E X AX y just follow

n n

if it has the following prop erty if the partial game

the denition of I t

k

trees b eginning at two dierent OR no des are identi

When n is even to provethat I t is the greatest

n

cal then the winning strategy picks the same AN D

xp oint let y b e a game tree We showthat

successor at these two OR no des A historyfree win

I t Weproveby induction on k that is a game

n

ning strategy for Player I I is dened similarly

tree with a strategy for player I st

Theorem If a player A II I has a winning

AFt ev en s s s s i s is

n  i k  i

strategy in a parity innite game then A has a history

col or and at s E X AX y

n k 

least mu is picked while the rst inequality follows free winning strategy

b ecause x has col or n k

j n

A partial game tree b eginning at no de x of a game



Fact x x tree will b e denoted A partial winning strategy

k   k  x



and the tree dened by it b eginning at no de x in the

This follows by denition of noting that

k 

winning strategy will b e denoted by

x

x

k  

Proof Weprove the result for A b eing Player I The

Thus x x

k   k  i

case of Player I I is handled similarlyby switching the

Determinacy of Online games

OR and the AN D no des

Consider an innite game tree in whicheach OR

Supp ose player I has a winning strategy As

no de is colored with col or where n is a natural num

n

remarked earlier the winning strategy denes a tree

b er RS construct a game to study comp etitive

in the game tree Consider the following maps

k 

ness of online algorithms in which Player I wins a

from the no des in to the class of all ordinals

particular playsay x x OR iik x has

 k

x if along all paths in if there is a

k  x

col or ni

n

no de lab elled col or then there is a no de b efore it

k 

lab elled col or where nk

As b efore let I b e the set of game trees in which

n

x x supf y there is a

player I has a winning strategyand II b e the set of

k   k  k 

path from x to y such that y is lab elled col or and

game trees in whichplayer I I has a winning strategy

k 

there is no no de in this path with col or nk g

Then aproofmuch simpler than Theorem gives

n

where x isif x is lab elled col or else

the following proving determinacy of this game

k  k 

The ab ove denition is welldened b ecause along

Theorem

all paths for all k every o ccurence of col or is

I y i xEX AXx col or n i y

k 

n

followed by a col or n k Thus the ab ove

II yixAX E X x col or n i y

n

n

inductive denition is wellfounded

This simplies the existential pro of in RS

Let with a leftlexicographic

of derandomization of comp etitive online algorithms

m m 

ordering whichisagainawellordering Welet

for task systems where RS was invoking Wolfes

k 

The historyfree strategy we

pro of W of determinacy of F games Infact a

m m k 

dene essentially picks at each ORno de xtheAN D

similar MuCalculus characterization can b e given for

no de with the least in case of contention ie if

F countable union of closed sets games elucidating

in there are other OR no des with partial game

Wolfes pro of We use terminology from Mo

S

trees identical to with picking dierent AN D

x

Let A F b e the winning set for Player I

i

i

successors at these OR no des

X where each F is closed in the pro duct top ology of

i



Wenowshowthat the strategy so obtained is

It is well known Mo that a closed set F is exactly

i



indeed a winning strategy for Player I Supp ose in

the innite paths in a Tree T on X Wesaythat T

i i

there is a path suc h that eventually col or app ears

holds on the last no de of a path p in the game tree i

k 

innitely often and no col or n k app ears

p o ccurs in T

n

i

Weshow that with every two such o ccurences x and



Theorem I y i xE X AX x T y and

i

x of col or decreases thus contradicting that

i k 

II yixAX E X x T y

i

isawellorder Here x is the successor of x in

j j 

 

Acknowledgement

Let the successor of x in be x

j 

j



We thank Jo e Halp ern and Prasad Sistla for p oint

Fact j i j x x

k  j k 

j 

ing out the alternative argument establishing equiva

x

k  j 

lence of the MuCalculus and tree automata by trans



The second inequality follows b ecause in the lation through SnS

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